On the convergence rates of discrete solutions to the Wave Kinetic Equation

Michele Dolce; Ricardo Grande; Michele Dolce; Ricardo Grande

doi:10.3934/mine.2024022

Mathematics in Engineering

2024, Volume 6, Issue 4: 536-558. doi: 10.3934/mine.2024022

Previous Article Next Article

Research article Special Issues

On the convergence rates of discrete solutions to the Wave Kinetic Equation

Michele Dolce ^{1
,
,},
Ricardo Grande ²

1.
Institute of Mathematics, EPFL, Station 8, Lausanne 1015, Switzerland
2.
International School for Advanced Studies (SISSA), Via Bonomea 265, Trieste 34136, Italy

Received: 22 April 2024 Revised: 24 July 2024 Accepted: 25 July 2024 Published: 01 August 2024

In this paper, we consider the long-term behavior of some special solutions to the Wave Kinetic Equation. This equation provides a mesoscopic description of wave systems interacting nonlinearly via the cubic NLS equation. Escobedo and Velázquez showed that, starting with initial data given by countably many Dirac masses, solutions remain a linear combination of countably many Dirac masses at all times. Moreover, there is convergence to a single Dirac mass at long times. The first goal of this paper is to give quantitative rates for the speed of said convergence. In order to study the optimality of the bounds we obtain, we introduce and analyze a toy model accounting only for the leading order quadratic interactions.
- Wave Kinetic Equation,
- wave turbulence,
- asymptotic behavior
Citation: Michele Dolce, Ricardo Grande. On the convergence rates of discrete solutions to the Wave Kinetic Equation[J]. Mathematics in Engineering, 2024, 6(4): 536-558. doi: 10.3934/mine.2024022

Related Papers:

Abstract

In this paper, we consider the long-term behavior of some special solutions to the Wave Kinetic Equation. This equation provides a mesoscopic description of wave systems interacting nonlinearly via the cubic NLS equation. Escobedo and Velázquez showed that, starting with initial data given by countably many Dirac masses, solutions remain a linear combination of countably many Dirac masses at all times. Moreover, there is convergence to a single Dirac mass at long times. The first goal of this paper is to give quantitative rates for the speed of said convergence. In order to study the optimality of the bounds we obtain, we introduce and analyze a toy model accounting only for the leading order quadratic interactions.

References

[1]	T. Buckmaster, P. Germain, Z. Hani, J. Shatah, Onset of the wave turbulence description of the longtime behavior of the nonlinear Schrödinger equation, Invent. Math., 225 (2021), 787–855. http://dx.doi.org/10.1007/s00222-021-01039-z doi: 10.1007/s00222-021-01039-z
[2]	C. Collot, P. Germain, On the derivation of the homogeneous kinetic wave equation, unpublished work.
[3]	Y. Deng, Z. Hani, On the derivation of the Wave Kinetic Equation for NLS, Forum Math., Pi, 9 (2021), e6. http://dx.doi.org/10.1017/fmp.2021.6 doi: 10.1017/fmp.2021.6
[4]	Y. Deng, Z. Hani, Propagation of chaos and the higher order statistics in the wave kinetic theory, J. Eur. Math. Soc., 2024. http://dx.doi.org/10.4171/jems/1488
[5]	Y. Deng, Z. Hani, Derivation of the Wave Kinetic Equation: full range of scaling laws, arXiv, 2023. http://dx.doi.org/10.48550/arXiv.2301.07063
[6]	Y. Deng, Z. Hani, Full derivation of the Wave Kinetic Equation, Invent. math., 233 (2023), 543–724. http://dx.doi.org/10.1007/s00222-023-01189-2 doi: 10.1007/s00222-023-01189-2
[7]	Y. Deng, Z. Hani, Long time justification of wave turbulence theory, arXiv, 2023. http://dx.doi.org/10.48550/arXiv.2311.10082
[8]	M. Escobedo, J. J. L. Velázquez, On the theory of weak turbulence for the nonlinear Schrödinger equation, Mem. Amer. Math. Soc., 238 (2015), 1124. http://dx.doi.org/10.1090/memo/1124 doi: 10.1090/memo/1124
[9]	P. Germain, A. Ionescu, M. B. Tran, Optimal local well-posedness theory for the kinetic wave equation, J. Funct. Anal., 279 (2020), 108570. http://dx.doi.org/10.1016/j.jfa.2020.108570 doi: 10.1016/j.jfa.2020.108570
[10]	A. Hannani, M. Rosenzweig, G. Staffilani, M. B. Tran, On the wave turbulence theory for a stochastic KdV type equation–Generalization for the inhomogeneous kinetic limit, arXiv, 2022. http://dx.doi.org/10.48550/arXiv.2210.17445
[11]	K. Hasselmann, On the non-linear energy transfer in a gravity-wave spectrum. Ⅰ. General theory, J. Fluid Mech., 12 (1962), 481–500. http://dx.doi.org/10.1017/S0022112062000373 doi: 10.1017/S0022112062000373
[12]	K. Hasselmann, On the non-linear energy transfer in a gravity wave spectrum. Ⅱ. Conservation theorems; wave-particle analogy; irreversibility, J. Fluid Mech., 15 (1963), 273–281. http://dx.doi.org/10.1017/S0022112063000239 doi: 10.1017/S0022112063000239
[13]	A. H. M. Kierkels, J. J. L. Velázquez, On the transfer of energy towards infinity in the theory of weak turbulence for the nonlinear Schrödinger equation, J. Stat. Phys., 159 (2015), 668–712. http://dx.doi.org/10.1007/s10955-015-1194-0 doi: 10.1007/s10955-015-1194-0
[14]	A. H. M. Kierkels, J. J. L. Velázquez, On self-similar solutions to a kinetic equation arising in weak turbulence theory for the nonlinear schrödinger equation, J. Stat. Phys., 163 (2016), 1350–1393. http://dx.doi.org/10.1007/s10955-016-1505-0 doi: 10.1007/s10955-016-1505-0
[15]	S. Nazarenko, Wave turbulence, Vol. 825, Springer Science & Business Media, 2011. http://dx.doi.org/10.1007/978-3-642-15942-8
[16]	L. Nordheim, On the kinetic method in the new statistics and application in the electron theory of conductivity, Proc. R. Soc. London. Ser. A, Containing Papers of a Mathematical and Physical Character, 119 (1928), 689–698. http://dx.doi.org/10.1098/rspa.1928.0126
[17]	R. Peierls, Zur kinetischen theorie der Wärmeleitung in Kristallen, Ann. Phys., 395 (1929), 1055–1101. http://dx.doi.org/10.1002/andp.19293950803 doi: 10.1002/andp.19293950803
[18]	G. Staffilani, M. B. Tran, On the wave turbulence theory for stochastic and random multidimensional KdV type equations, arXiv, 2021. http://dx.doi.org/10.48550/arXiv.2106.09819

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)